La Expresión sonora

The methods of Independent Replications (IR) overcome the analytical prob- lem created by the autocorrelated nature of the original output data in a conceptually simple way: the simulation is repeated a number of times, each time using dierent, independent sequence of random numbers. Observa- tions collected during a replication are used to obtain only one, secondary data point: the average value over observations collected during this replica- tion. The set of averages are used in further statistical analysis as evidently independent and identically distributed output data.

However, the average of the observations in each replication may be a strongly biased estimator of the steady state mean, because observations collected during the initial transient period of each replication may not be representative of observations from the process in its steady state. In the context of IR, we can identify three approaches for dealing with the problem of initial transient bias. The rst approach is to estimate the length of the initial transient period, and discard data collected during the transient pe- riod. The argument in support of this approach is that it should reduce the bias of the estimator, and hence improve its quality in terms of its coverage. The second approach is to ignore the presence of an initial transient period, and assume that all data were collected from the process in its steady state. The argument in support of this approach is that, by not discarding any observations, more observations are available for producing the estimate, so the variance of the estimator, and hence its relative precision should be im- proved. The third approach, developed by us, is to directly consider the bias in the estimator, and also refrain from discarding any observations. Accord- ing to this approach, either the interval estimate for the mean is adjusted to accounted for the estimated bias, or the length of each replication is in- creased until it has being determined that the biased caused by observations collected during the transient period is small, relative to the variance of the esitmator [YAU96].

In practically all reported cases when sequential versions of IR were ap- plied in conjunction with initial data deletion (the rst approach to dealing with the initial transient), the length of replications and the number of dis- carded initial observations had to be predicted in advance and kept constant, while the number of replications was adjusted dynamically.

Having generated kr replications, with m observations collected during

each of them, we have kr sequences of data, (x11, x12, ...,x1m), (x21, x22, ...,

x2m), ..., (xkr1,xkr2, ...,xkrm). Now n0 initial observations are removed from

each replication. The replication means are then calculated as : Yi =Xi(m,n 0i) = 1 m,n 0i m X j=n 0i +1 xij i = 1;2;:::;kr (2.9)

The secondary data, Y1, Y2, ... , Ykr, can be considered as realisations

of i.i.d. random variables that can be used to get the point and interval estimator of x by substituting n by kr, xi byY , and Xn by

=

XIR= 1k_r kr

X

i=1

Yi (2.10)

in Eqns. 2.1 to 2.4. Authors of such sequential simulations tried to nd the best trade-o between the number of replications and their length, for achieving good quality of the nal estimators. For example, at least 100 nal observations in each replication (having discarded initial observations) were suggested to secure normality of the replication means [FISH78, p.122]. Moreover, as shown in [LAW77] and [KELT84], it is better to keep replica- tions longer than to make more replications, since it would usually improve the nal coverage too.

In a fully sequential version of IR using discarding, both the length of initial transient period and the length of each replication should be deter- mined dynamically during simulation runs. The rst step in this direction has been recently proposed in [PAWL91], with dierent numbers of initial observations discarded during dierent replications (following a sequential, independent estimation of the length of initial transient length) but with a xed total replication length in steady state, i.e. having deleted initial obser- vations. Full automation of this method would additionally require making the length of each replication related to the dynamic characteristics of the simulated process. It could be done by making the length of each replication related to the length of its initial transient period. But, such an enhancement of IR creates a statistical problem. Namely, the variance of the global mean in such scenario would be a weighted sum of replication means calculated

over dierent sample sizes. It requires special precautions to be taken for avoiding worsening the quality of such global estimators. This eect has not been studied yet, neither theoretically nor empirically.

As mentioned, the second application of IR is to ignore the possibility that the initial observations of each replication may have being generated whilst the process was in a non-stationary state. The point and interval estimator under this approach are the same as that for IR with discarding described above, except thatn0i(the number of initial observations discarded

per replication) would be zero.

Since our initial results of the coverage analysis of IR with discarding in- dicated good quality of this method, [PAWL91], [PAWL92], we have included this technique in our comparative studies, the results of which are reported in Chapter 3. Unfortunately, IR does not seem to be a good candidate for sequential simulation under time constraints, for example when executed on single sequential computer, since it may require prohibitively long time of execution caused by discarding initial observations (collected during the ini- tial transient periods) at the beginning of each replication. It can lead to a poor utilization of the total simulation time if the lengths of initial transient periods are signicant.